Journal of Physics Conference Series

Solution of spherical equation in 3 dimensions for hydrogen atom with quantum numbers 4≤ n≤ 5

Bambang Supriadi, Zainur Rasyid Ridlo, F Fuadah, M A Halim, M Maulana, Y R Santoso

Published Apr 1, 2019• 2 citations

Abstract

The lightest and simplest arrangement of protons and electrons is a hydrogen atom. One of the wave characteristics of an atom can be known using the Schrodinger equation. Schrodinger's equations of spherical coordinates consist of radial equations and angular equations. Quantum numbers 4≤ n ≤ 5from the Schrodinger equation sphere coordinates produce different angular wave functions. To find out the angular wave equation using normalization conditions . Based on the normalization conditions for n = 4 a wave function is produced , while for n = 5 a wave function is .

DOI: 10.1088/1742-6596/1211/1/012054

Related papers

The function of the radial wave of a hydrogen atom in the principal quantum numbers (n) 4 and 5

Bambang Supriadi, Alex Harijanto, M Maulana +3 more

The simplest atom consisting of a proton and an electron is a hydrogen atom. The energy level and the probability of finding an electron in a hydrogen atom can be determined based on the function of radial wave. The function of radial wave of a hydrogen atom is influenced by the principal quantum number ( n ) and the orbital quantum number ( l ). Completion of the function of radial wave of a hydrogen atom in the principal quantum numbers( n ) 4 and 5 uses the time-independent Schrodinger equation approach in spherical coordinates, variable separation method and uses the associated Laguerre function. The normalized of the function of radial wave of a hydrogen atom R nl ( r ) at n = 4 produces four wave functions:R 40 ( r ); R 41 ( r ); R 42 ( r ); R 43 ( r ), whereas at n = 5 produces five wave functions:R 50 ( r ); R 51 ( r ); R 52 ( r ); R 53 ( r ); R 54 ( r ). The results of this research in the form of the functions of radial wave of hydrogen atom can be used to determine the behavior of electron in a hydrogen atom.

Journal of Physics Conference SeriesApr 1, 20194 citations

Hydrogen-like Atomic Wavefunctions: A First Sketch

Jochen Autschbach

Abstract This chapter reiterates the quantum numbers for atomic orbitals, known from general chemistry, and places them into the context developed so far. It is sketched how the Schrodinger equation (SE) for the hydrogen atom hydrogen-like systems (one electron plus a nucleus of charge Z) is set up. When the nucleus is treated as a fixed point charge, the SE is only for the electron. The solutions of the SE can be obtained by switching to spherical polar coordinates, such that the variables are separable in terms of the electron distance from the nucleus, r, and two angles. The kinetic energy of the electron then has a radial component, and an angular component. The latter is associated with the angular momentum quantum number, which is codified by the letters s, p, d, f, and so forth. A step by step solution of the SE is provided later, in chapter 19.

Oxford University Press eBooksDec 21, 20200 citations

Hydrogen-like Atom and Atomic Orbitals

Caio Lima Firme

This chapter physical properties of the hydrogen atom and its Hamiltonian function as a particle in a spherically symmetric potential having a wave function as a product of the radial function and spherical harmonic function. Hereafter, it comes to two independent wave equations: the radial equation and the angular equation. After some derivation, it is shown that the angular equation is a product of the azimuthal angle equation and polar angle equation (as discussed in the previous chapter). The azimuthal angle equation is similar to the particle in a ring problem from the previous chapter and after some derivation the polar angle equation becomes an associated Legendre equation with well-known polynomials as solutions. Next, the solution to the radial equation is presented which becomes after some derivation an associated Laguerre equation with well-known polynomials as solutions and from this result is obtained the energy equation for the hydrogen-like atom. Subsequently, it is derived the normalization constant of the radial function. Afterwards, the numerical analysis of the radial equation of the hydrogen atom along with its corresponding Fortran code is presented. Finally, the hydrogen atomic orbitals equations and plots are depicted followed by the fine structure of the hydrogen (198 words).

CRC Press eBooksMar 21, 20220 citations

FURTHER STUDY ON THE SOLUTION OF SCHRÖDINGER EQUATION OF HYDROGEN-LIKE ATOM

Zhong-Liang Zhang, Wu-Shou Zhang, Zhao-qun Zhang

In this work the Schrödinger equation of the hydrogen-like atom is analytically solved. Three sets of analytical solution are obtained if the factor r−l is not neglected. The first solution is the same as the traditional radial wave function; another one diverges; the last one is far different from the traditional solution. On the consideration of the finite size of the nucleus, the third wave function does not diverge while r approaches to zero. Its radial wave function has below characteristics: (1) the angular-momentum quantum number l must be greater than the principal quantum number n; (2) l must not be 0 or 1; (3) the electron-cloud distribution differs from the traditional one; (4) the electron is closer to the nucleus by comparison with that in traditional results. On the other hand, the validity of solutions needs to be verified experimentally. 1.

Aug 25, 20150 citations

Bound States in Three Dimensions: The Atom

Duncan G. Steel

Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.

Apr 30, 20210 citations

EXPONENTIALLY DECAYING WAVE FUNCTION AND QUANTIZED ENERGY FOR NANO SPHERICAL PARTICLES AND HYDROGEN LIKE ATOMS

Ghada E. S. Elammeen Amna E. Musa

The solution of radial Schrödinger equation for hydrogen like atoms is complex. One needs simple solution the behavior of nano particle needs also to be described. Nano spherical particles are isolated non interacting particles that obey quantum laws. Thus one expects their behavior to resemble that of particles in a box, or hydrogen like atoms. The spherical nano shape requires using spherical coordinate to solve Schrödinger Equation. The solution found that the wave function is diminishing outside the sphere, which agrees with observations. This solution is simpler than the conventional one, where it is a sum of sine and cosine function.

Mar 16, 20190 citations